SCIENCE · INVESTIGATE AND EXPLAIN

Use an integral law without inventing a constant field

Two infinite parallel wires need not produce a field with circular symmetry about their midpoint. Put them at x=−0.4 and +0.4 m, with signed currents +2 A and a selectable second current along z. A centered circular path of radius at least 1 m encloses both wires, but is centered on neither individual field pattern.

Learning outcome: Compare second currents +2,0,−2 A. Record exact circulation, numerical circulation and fields at the rightmost and top points. Explain which symmetry would be required to replace the entire integral by a single measured B times circumference.

Starting recap

Review cross products, source integration measures, line integrals, differential initial states and signed work. Distinguish symmetry premises from integral laws.

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Try

With currents +2 A and −2 A and path radius 1 m, total circulation is zero. At the top point (0,1), however, the horizontal components cancel while the vertical component is 0.32/1.16 µT≈0.275862 µT. That is radial relative to the chosen path, so a zero circulation does not erase the local field.

Open the experiment

Predict

If equal opposite enclosed currents give zero circulation, must local field vanish?

Your prediction and reason

Experiment

Compare second currents +2,0,−2 A. Record exact circulation, numerical circulation and fields at the rightmost and top points. Explain which symmetry would be required to replace the entire integral by a single measured B times circumference.

Test circulation without pretending the field is constant

Worked teaching model · synthetic values · Two infinite z-directed wires at x=−.4 and +.4 m carry +2 A and a A. Circular paths centered at origin enclose both and never intersect wires. The field on the path is not generally constant or purely tangent. A 240-point midpoint line integral is a numerical approximation, not an extra symmetry premise.

Second signed current: -1 A Circular integration radius: 1 m

Reset keeps your prediction, captures and practice history.

Tangential field varies around an enclosing path

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: two offset wires (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Tangential field varies around an enclosing path

two offset wires

polar angle (rad)	tangential magnetic field (T)
0	-4.76190e-8
0.07853982	-4.59235e-8
0.1570796	-4.09723e-8
0.2356194	-3.31434e-8
0.3141593	-2.29832e-8
0.3926991	-1.11113e-8
0.4712389	1.86739e-9
0.5497787	1.54262e-8
0.6283185	2.91499e-8
0.7068583	4.27417e-8
0.7853982	5.60133e-8
0.863938	6.88668e-8
0.9424778	8.12737e-8
1.021018	9.32566e-8
1.099557	1.04874e-7
1.178097	1.16209e-7
1.256637	1.27359e-7
1.335177	1.38435e-7
1.413717	1.49553e-7
1.492257	1.60837e-7
1.570796	1.72414e-7
1.649336	1.84416e-7
1.727876	1.96980e-7
1.806416	2.10247e-7
1.884956	2.24359e-7
1.963495	2.39462e-7
2.042035	2.55694e-7
2.120575	2.73183e-7
2.199115	2.92032e-7
2.277655	3.12306e-7
2.356194	3.34002e-7
2.434734	3.57024e-7
2.513274	3.81141e-7
2.591814	4.05947e-7
2.670354	4.30823e-7
2.748894	4.54917e-7
2.827433	4.77159e-7
2.905973	4.96325e-7
2.984513	5.11165e-7
3.063053	5.20580e-7
3.141593	5.23810e-7
3.220132	5.20580e-7
3.298672	5.11165e-7
3.377212	4.96325e-7
3.455752	4.77159e-7
3.534292	4.54917e-7
3.612832	4.30823e-7
3.691371	4.05947e-7
3.769911	3.81141e-7
3.848451	3.57024e-7
3.926991	3.34002e-7
4.005531	3.12306e-7
4.08407	2.92032e-7
4.16261	2.73183e-7
4.24115	2.55694e-7
4.31969	2.39462e-7
4.39823	2.24359e-7
4.47677	2.10247e-7
4.555309	1.96980e-7
4.633849	1.84416e-7
4.712389	1.72414e-7
4.790929	1.60837e-7
4.869469	1.49553e-7
4.948008	1.38435e-7
5.026548	1.27359e-7
5.105088	1.16209e-7
5.183628	1.04874e-7
5.262168	9.32566e-8
5.340708	8.12737e-8
5.419247	6.88668e-8
5.497787	5.60133e-8
5.576327	4.27417e-8
5.654867	2.91499e-8
5.733407	1.54262e-8
5.811946	1.86739e-9
5.890486	-1.11113e-8
5.969026	-2.29832e-8
6.047566	-3.31434e-8
6.126106	-4.09723e-8
6.204645	-4.59235e-8
6.283185	-4.76190e-8

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Enclosed signed current	1	A
Exact Ampere circulation	1.25664e-6	T m
Midpoint numerical circulation	1.25664e-6	T m
Circulation residual	2.11758e-22	T m
Field magnitude at rightmost path point	4.76190e-8	T
Field magnitude at top path point	2.69319e-7	T
Radial component at top point	2.06897e-7	T

An integral law remains true even when the chosen path lacks enough symmetry to extract one local field value.

Each infinite wire contributes mu0 I/(2pi rho) azimuthally around its own axis. Their sum is not generally tangent to, or constant on, a circle centered between them. Integrate B dot dl rather than multiplying one sampled magnitude by circumference. Opposite equal currents give zero total circulation but can leave nonzero local field. The grid checks the line integral of the stipulated infinite-wire superposition, not a finite apparatus.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

An integral law remains true even when the chosen path lacks enough symmetry to extract one local field value.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Specify sources, directions and independent premises

Derive the integral or vector evolution

For a wire at x0, its planar field is proportional to [−y,x−x0]/[(x−x0)²+y²], multiplied by its signed current. Add these vectors before taking a tangent component. The circulation is the integral of that tangent component times path length; it is not generally one sampled field magnitude times the circumference.

Check energy, symmetry and evidence limits

The numerical midpoint line integral should approach μ0 times the sum of enclosed signed currents. Equal opposite currents make that integral zero while leaving nonzero field at many points. A radial field component relative to the chosen midpoint circle is also possible. The integral law is not failing; the extra constant-tangent-field assumption would be failing.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

For enclosed currents +2 A and +1 A, find circulation in microtesla metres using the rounded μ0.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original derivations and synthetic numerical evidence checked 31 October 2026 against the primary explanation. The rounded teaching permeability is not asserted to be an exact modern SI constant; NIST CODATA 2022 lists its measured uncertainty. No live electrical construction, beam, strong-magnet handling, medical or physical certification claim is requested. Public feedback remains separate from retained first work, critique, revision and fresh assigned evidence.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which source geometry, symmetry, initial state or calibration premise is necessary, and what independent observation could reject it?

Your explanation and limitations

Continue

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Previous: Enclose a nonuniform current and its return path Next: Integrate magnetic rotation and parallel translation

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